11.1-11.2 binomial theorem & binomial expansion. pascal’s triangle

11.1-11.2 Binomial 11.1-11.2 Binomial Theorem & Binomial Theorem & Binomial Expansion Expansion

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Page 1: 11.1-11.2 Binomial Theorem & Binomial Expansion. Pascal’s Triangle

11.1-11.2 Binomial Theorem & 11.1-11.2 Binomial Theorem & Binomial ExpansionBinomial Expansion

Pascal’s Triangle

Pascal’s Triangle with even and odd numbers colored differently:

Page 4: 11.1-11.2 Binomial Theorem & Binomial Expansion. Pascal’s Triangle

Expanding Binomials…

What pattern(s) do you see?.....

543223455

4322344

32233

222

510105 )(

464 )(

33 )(

2 )(

)(

1 )(

babbababaaba

babbabaaba

babbaaba

bababa

baba

Page 5: 11.1-11.2 Binomial Theorem & Binomial Expansion. Pascal’s Triangle

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

Use this triangle to expand binomials of the form (a+b)n. Each row corresponds to a whole number n. The first row consists of the coefficients of (a+b)n when n = 0.

Example 1

Example 2

Page 6: 11.1-11.2 Binomial Theorem & Binomial Expansion. Pascal’s Triangle

Expand (a + b)6

• 1 6 15 20 15 6 1 (coefficients from Pascal’s triangle)

• 1a6 6a5 15a4 20a3 15a2 6a1 1a0 (exponents of a begin with 6 and decrease)

• 1a6b0 6a5b1 15a4b2 20a3b3 15a2b4 6a1b5 1a0b6

(exponents of b begin with 0 and increase by 1)• a6 + 6a5b + 15a4b2 + 20a3b3 + 15a2b4 + 6ab5 + b6

(expansion in standard, simplified form)

Return to Triangle

Page 7: 11.1-11.2 Binomial Theorem & Binomial Expansion. Pascal’s Triangle

Expand (x – 2)3

• 1 3 3 1 (coefficients from Pascal’s triangle)• 1a3 3a2 3a1 1a0 • 1a3b0 3a2b1 3a1b2 1a0b3

• Now substitute x for a and –2 for b:• 1(x)3(–2)0 3(x)2(–2)1 3(x)1(–2)2 1(x)0(–2)3

• x3 – 6x2 + 12x – 8 (expansion in standard, simplified form)

Return to Triangle

(let a = x and b = –2)

Page 8: 11.1-11.2 Binomial Theorem & Binomial Expansion. Pascal’s Triangle

Factorial Notation

! ( 1)( 2)( 3).......(3)(2)(1)n n n n n

7! 7(6)(5)(4)(3)(2)(1)

4! (4)(3)(25 5 )( ) )(1

Read as “n factorial”

0! 1

8(7)(6!)

6! 8(7) 56

Page 9: 11.1-11.2 Binomial Theorem & Binomial Expansion. Pascal’s Triangle

Finding a particular term in a Binomial Expansion:

( )! !n r rn

a bn r r

6 3

! !

9 8 7 6

6 3 2 1

( )( )( !)

!( )( )( )

Ex.: Find the 4th term in expansion of (a + b)9:

This is the 4th term, so value of r (b’s exponent) is 4 – 1 = 3.

This means the exponent for a is 9 – 3, or 6.

So, we have the variables of the 4th term: a6b3

Formula for finding a particular term in expansion of (a + b)n is:

Coefficient is: 6 384a b

Page 10: 11.1-11.2 Binomial Theorem & Binomial Expansion. Pascal’s Triangle

Finding a particular term in a Binomial Expansion:

( )! !n r rn

a bn r r

5 7

! !

12 11 10 9 8 7

5 4 3 2 7

( )( )( )( )( !)

( )( )( )( !)

Ex.: Find the 8th term in expansion of (2x – y)12:

This is 8th term, exponent for b is 8 – 1 = 7.

This means the exponent for a is 12 – 7, or 5.

So, the variables of the 8th term: a5b7,

Formula for finding a particular term in expansion of (a + b)n is:

Coefficient: 5 7792 32x y ( )( )

or (2x)5(–y)7.

5 725344x y

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